Question

Find the derivative of f(x) 5/x + 7/x^2​

Answer 1

`\rm \: f(x) = (5)/(x) + \frac{7}{ {x}^(2) }`

Differentiating both sides with respect to x

`\rm (d)/(dx) ( {f}( x) = (d)/(dx) \bigg( (5)/(x) + \frac{7}{ {x}^(2) } \bigg)`

Using u + v rule

`\rm \: {f}^( \prime) x = (d)/(dx) \bigg( (5)/(x) \bigg) + (d)/(dx) \bigg( \frac{7}{ {x}^(2) } \bigg)`

`\rm \: {f}^( \prime) x = 5. (d)/(dx) ( {x}^( - 1) ) + 7. (d)/(dx) ( {x}^( - 2) )`

`\rm \: {f}^( \prime) x = 5.( - 1. {x})^( (- 1 - 1)) + 7.( - 2. {x})^( - 2 - 1)`

`\rm \: {f}^( \prime) x = { - 5x}^( - 2) { - 14x}^( - 3)`

`\rm \: {f}^( \prime) x = - \frac{5}{ {x}^(2) } - \frac{14}{ {x}^(3) }`

`\rm \: {f}^( \prime) x = - \bigg(\frac{5}{ {x}^(2) } + \frac{14}{ {x}^(3) } \bigg)`

Hense The required Derivative is answered.

Derivative Formulae:-

`\boxed{\begin{array}c \rm \: \underline{function}& \rm \underline{Derivative} \\ \\ \rm (d)/(dx) ({x}^(n)) \: \: \: \: \: \: \: \: \: \ & \rm nx^(n-1) \\ \\ \rm \: (d)/(dx)(constant) &0 \\ \\ \rm (d)/(dx)( \sin x )\: \: \: \: \: \: & \rm \cos x \\ \\ \rm (d)/(dx)( \cos x ) \: \: \: & \rm - \sin x \\ \\ \rm (d)/(dx)( \tan x ) & \rm \: { \sec}^(2)x \\ \\ \rm (d)/(dx)( \cot x ) & \rm- { \csc }^(2)x \\ \\ \rm (d)/(dx)( \sec x ) & \rm \sec x. \tan x \\ \\\rm (d)/(dx)( \csc x ) & \rm \: - \csc x. \cot x\\ \\ \rm (d)/(dx)(x) \: \: \: \: \: \: \: & 1 \end{array}}`